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introduction to probability statistics

Introduction To Probability And Statistics 3rd Edition William Mendenhall - Solutions

A laboratory test for a rare disease affecting 2% of the population is either positive, indicating the rare disease is present, or negative, indicating the disease is not present. However, when people who have the rare disease are tested in the laboratory, 90% of the tests results were positive and
Canada has a Road Safety Vision of having the safest roads in the world. Yet, the lead- ing cause of death of Canadian children remains vehicle crashes. In 2006, a national child seat safety survey was conducted by an AUTO21 research team in collaboration with Transport Canada to empirically
In his exciting novel Congo, Michael Crichton describes a search by Earth Resources Technology Service (ERTS), a geological survey company, for deposits of boron- coated blue diamonds, diamonds that ERTS believes to be the key to a new generation of optical computers.13 In the novel, ERTS is racing
4.139 Bringing Home the Bacon The following information reflects the results of a survey reported by Mya Frazier in an Ad Age Insights white paper. 12 Working spouses were asked "Who is the household breadwinner?" Suppose that one person is selected at random from these 200 individuals.a. What is
4.138 Independence and Mutually Exclusive Suppose that P(A)=0.3 and P(B)=0.4.a. If P(AB)=0.12, are A and B independent? Justify your answer.b. If P(AUB)=0.7, what is P(AB)? Justify your answer.c. If A and B are independent, what is P(A|B)?d. If A and B are mutually exclusive, what is P(A|B)?
4.137 Orchestra Politics The board of directors of a major symphony orchestra has voted to create a players' committee for the purpose of handling employee complaints. The council will consist of the president and vice-president of the symphony board and two orchestra representatives. The two
4.136 Viruses A certain virus afflicted the families in three adjacent houses in a row of 12 houses. If three houses were randomly chosen from a row of 12 houses, what is the probability that the three houses would be adjacent? Is there reason to believe that this virus is contagious?
4.135 MRIs An article in The American Journal of Sports Medicine compared the results of magnetic resonance imaging (MRI) evaluation with arthroscopic surgical evaluation of cartilage tears at two sites in the knees of 35 patients. The 2 x 35 = 70 examina- tions produced the classifications shown
4.134 Pepsi or Coke? A taste-testing experiment is conducted at a local supermarket, where passing shoppers are asked to taste two soft-drink samples- one Pepsi and one Coke and state their preference. Suppose that four shoppers are chosen at random and asked to participate in the experiment, and
4.133 A Colour Recognition Experiment An experiment is run as follows-the colours red, yellow, and blue are each flashed on a screen for a short period of time. A subject views the colours and is asked to choose which one was flashed for the longest time. The experiment is repeated three times with
4.132 Flextime The number of companies offering flexible work schedules has increased as companies try to help employees cope with the demands of home and work. One flextime schedule is to work four 10-hour shifts. A survey provided the following information for 220 firms located in two cities in
4.131 Tossing a Coin How many times should a coin be tossed to obtain a probability equal to or greater than 0.9 of observing at least one head?
4.130 Blood Tests To reduce the cost of detecting a disease, blood tests are conducted on a pooled sample of blood collected from a group of n people. If no indication of the disease is present in the pooled blood sample (as is usually the case), none have the disease. If analysis of the pooled
4.129 Blood Pressure Meds A research physician compared the effectiveness of two blood pressure drugs A and B by administering the two drugs to each of four pairs of identical twins. Drug A was given to one member of a pair; drug B to the other. If, in fact, there is no difference in the effects of
4.128 Mass Transit Only 40% of all people in a community favour the development of a mass transit system. If four citizens are selected at random from the community, what is the probability that all four favour the mass transit system? That none favours the mass transit system?
4.127 Quality Control A quality-control plan calls for accepting a large lot of crankshaft bearings if a sample of seven is drawn and none are defective. What is the probability of accepting the lot if none in the lot are defective? If 1/10 are defective? If 1/2 are defective?
4.126 Waiting Times Suppose that at a particular supermarket the probability of waiting 5 minutes or longer for checkout at the cashier's counter is 0.2. On a given day, a husband and wife decide to shop individually at the market, each checking out at differ- ent cashier counters. They both reach
4.125 Accident Insurance Accident records col- lected by an automobile insurance company give the following information: The probability that an insured driver has an automobile accident is 0.15; if an acci- dent has occurred, the damage to the vehicle amounts to 20% of its market value with
4.124 Work Tenure Suppose the probability of remaining with a particular company 10 years or longer is 1/6. A man and a woman start work at the company on the same day.a. What is the probability that the man will work there less than 10 years?b. What is the probability that both the man and the
4.123 Contract Negotiations Experience has shown that, 50% of the time, a particular union- management contract negotiation led to a contract settlement within a two-week period, 60% of the time the union strike fund was adequate to support a strike, and 30% of the time both conditions were
4.122 The Match Game Two people each toss a coin. They obtain a "match" if either both coins are heads or both are tails. Suppose the tossing is repeated three times.a. What is the probability of three matches?b. What is the probability that all six tosses (three for each man) result in tails?c.
4.121 Winning the Lottery II On Thursday, September 28, 2006, an 85-year-old lady in Etobicoke, Ontario, woke from a dream about winning a lottery, and wrote down the six numbers on a piece of paper. She bought one Lotto 6/49 lottery ticket that day using the numbers 1, 10, 18, 24, 31, and 46. On
4.120 Winning the Lottery On Wednesday, October 26, 2005, a single winning Lotto 6/49 lottery ticket was sold in Camrose, Alberta. The ticket belonged to 17 oil industry workers. According to the Alberta Lottery Fund, the winning numbers were 5, 11, 20, 30, 37, and 43. The prize was $54,000,000.a.
4.119 Rental Trucks A rental truck agency services its vehicles on a regular basis, routinely checking for mechanical problems. Suppose that the agency has six moving vans, two of which need to have new brakes. During a routine check, the vans are tested one at a time.a. What is the probability
4.118 Guided Missiles The failure rate for a guided missile control system is 1 in 1000. Suppose that a duplicate, but completely independent, control system is installed in each missile so that, if the first fails, the second can take over. The reliability of a missile is the probability that it
4.117 Bus or Subway A man takes either a bus or the subway to work with probabilities 0.3 and 0.7, respectively. When he takes the bus, he is late 30% of the days. When he takes the subway, he is late 20% of the days. If the man is late for work on a particular day, what is the probability that he
4.116 A Reticent Salesman A salesperson figures that the probability of consummating a sale during the first contact with a client is 0.4 but improves to 0.55 on the second contact if the client did not buy during the first contact. Suppose this salesperson makes one and only one callback to any
4.115 Racial Bias? Four union members, two from a minority group, are assigned to four distinctly differ- ent one-person jobs, which can be ranked in order of desirability.a. Define the experiment.b. List the simple events in S.c. If the assignment to the jobs is unbiasedthat is, if any one
4.114 Profitable Stocks An investor has the option of investing in three of five recommended stocks. Only two will show a substantial profit within the next five years, though it is unknown which ones. If the investor selects the three stocks at random (giving every com- bination of three stocks an
4.113 Plant Genetics Refer to the experiment con- ducted by Gregor Mendel in Exercise 4.65. Suppose you are interested in following two independent traits in snap peas-seed texture (S= smooth, s = wrinkled) and seed colour (Y=yellow, y = green)in a second- generation cross of heterozygous parents.
4.112 Catching a Cold Is your chance of getting a cold influenced by the number of social contacts you have? A study by Sheldon Cohen, a psychology profes- sor at Carnegie Mellon University, seems to show that the more social relationships you have, the less sus- ceptible you are to colds. A group
4.111 Fire Alarms A fire-detection device uses three temperature-sensitive cells acting independently of one another in such a manner that any one or more can activate the alarm. Each cell has a probability p=0.8 of activating the alarm when the tempera- ture reaches 38C or higher. Let x equal the
4.110 Fire Insurance A county containing a large number of rural homes is thought to have 60% of those homes insured against fire. Four rural homeowners are chosen at random from the entire population, and x are found to be insured against fire. Find the probability distribution for x. What is the
4.109 Heavy Equipment A heavy-equipment salesperson can contact either one or two customers per day with probability 1/3 and 2/3, respectively. Each contact will result in either no sale or a $50,000 sale with probability 9/10 and 1/10, respectively. What is the expected value of the daily sales?
4.108 Boxcars A boxcar contains seven com- plex electronic systems. Unknown to the purchaser, three are defective. Two of the seven are selected for thorough testing and are then classified as defective or non-defective. What is the probability that no defec- tives are found?
4.107 DVRs A retailer sells two styles of high-priced digital video recorders (DVR) that experience indicates are in equal demand. (Fifty percent of all potential cus- tomers prefer style 1, and 50% favour style 2.) If the retailer stocks four of each, what is the probability that the first four
4.106 Refer to Exercise 4.105. By summing the prob- abilities of simple events, find P(A), P(B), P(AB), P(AUB), P(C), P(ANC), and P(AUC).
4.105 Aspirin Two cold tablets are accidentally placed in a box containing two aspirin tablets. The four tablets are identical in appearance. One tablet is selected at random from the box and is swallowed by the first patient. A tablet is then selected at random from the three remaining tablets and
4.104 Whistle Blowers "Whistle blowers" is the name given to employees who report corporate fraud, theft, and other unethical and perhaps criminal activities by fellow employees or by their employer. Although there is legal protection for whistle blowers, it has been reported that approximately 23%
4.103 Playing the Slots A slot machine has three slots; each will show a cherry, a lemon, a star, or a bar when spun. The player wins if all three slots show the same three items. If each of the four items is equally likely to appear on a given spin, what is your probabil- ity of winning?
Suppose you have this general probability distribution:1. Enter the values of x and p(x) into columns C1 and C2 of a new MINITAB worksheet. In the gray boxes just below C3, C4, and C5, respectively, type the names "Mean," "Variance," and "Std Dev." You can now use the Calc Calculator command to
Suppose you have this general discrete probability distribution: X 0 1 3 3 5 0.15 p(x) 0.25 0.35 0.25
4.102 Actuaries A manufacturing representative is considering taking out an insurance policy to cover possible losses incurred by marketing a new product. If the product is a complete failure, the representative feels that a loss of $80,000 would be incurred; if it is only moderately successful, a
4.101 Shipping Charges From experience, a ship- ping company knows that the cost of delivering a small package within 24 hours is $14.80. The company charges $15.50 for shipment but guarantees to refund the charge if delivery is not made within 24 hours. If the com- pany fails to deliver only 2% of
4.100 No Time for Vegetables In a survey conducted by Ipsos Canada, 41% Canadians say that because of their busy lifestyles, they find it hard to prepare and eat enough vegetables. Suppose you had conducted your own telephone survey at the same time. You randomly called people and asked them
4.99 Health Canada Testing The maximum patent life for a new drug is, in most cases, 20 years. Sub- tracting the length of time required by Health Canada for testing and approval of the drug provides the actual patent life of the drug-that is, the length of time that a company has to recover
4.98 Insuring Your Diamonds You can insure a $50,000 diamond for its total value by paying a pre- mium of D dollars. If the probability of theft in a given year is estimated to be 0.01, what premium should the insurance company charge if it wants the expected gain to equal $1000?
4.97 The PGA One professional golfer plays best on short-distance holes. Experience has shown that the numbers x of shots required for 3-, 4-, and 5-par holes have the probability distributions shown in the table:What is the golfer's expected score on these holes?a. A par-3 holeb. A par-4 holec. A
4.96 Tennis, again The probability that a tennis player A can win a set from tennis player B is one measure of the comparative abilities of the two players. In Exercise 4.95 you found the probability distribution for X, the number of sets required to play a best-of-five- sets match, given that the
4.95 Tennis, Anyone? Two tennis professionals, A and B, are scheduled to play a match; the winner is the first player to win three sets in a total that cannot exceed five sets. The event that A wins any one set is independent of the event that A wins any other, and the probability that A wins any
4.94 Drilling Oil Wells Past experience has shown that, on the average, only 1 in 10 wells drilled hits oil. Let X be the number of drillings until the first success (oil is struck). Assume that the drillings represent independent events.a. Find p(1), p(2), and p(3).b. Give a formula for p(x).c.
4.93 Defective Equipment A piece of electronic equipment contains six computer chips, two of which are defective. Three chips are selected at random, removed from the piece of equipment, and inspected. Let X equal the number of defectives observed, where x = 0, 1, or 2. Find the probability
4.92 Gender Bias? A company has five applicants for two positions: two women and three men. Suppose that the five applicants are equally qualified and that no pref- erence is given for choosing either gender. Let x equal the number of women chosen to fill the two positions.a. Find p(x).b. Construct
4.91 Roulette Exercise 4.10 described the game of roulette. Suppose you bet $5 on a single numbersay, the number 18. The payoff on this type of bet is usually 35 to 1. What is your expected gain?
4.90 Which Key Fits? A key ring contains four office keys that are identical in appearance, but only one will open your office door. Suppose you ran- domly select one key and try it. If it does not fit, you randomly select one of the three remaining keys. If it does not fit, you randomly select one
4.89 Letterman or Leno? Who is the king of late night TV? An Internet survey estimates that, when given a choice between David Letterman and Jay Leno, 52% of the population prefers to watch Jay Leno. Suppose that you randomly select three late night TV watchers and ask them which of the two talk
4.88 Grocery Visits Let X represent the number of times a customer visits a grocery store in a one-week period. Assume this is the probability distribution of X:Find the expected value of X, the average number of times a customer visits the store. X 0 1 2 3 pix 0.1 0.4 0.4 0.1
4.87 Dice Let X equal the number observed on the throw of a single balanced die.a. Find and graph the probability distribution for X.b. What is the average or expected value of X?c. What is the standard deviation of X?d. Locate the interval 20 on the x-axis of the graph in parta. What proportion of
4.86 Probability Distribution II A random variable X can assume five values: 0, 1, 2, 3, 4. A portion of the probability distribution is shown here:a. Find p(3).b. Construct a probability histogram for p(x).c. Calculate the population mean, variance, and standard deviation.d. What is the
4.85 Probability Distribution | A random variable X has this probability distribution:a. Find p(4).b. Construct a probability histogram to describe p(x).c. Find , or, and or.d. Locate the interval 20 on the x-axis of the histo- gram. What is the probability that X will fall into this interval?e. If
4.84 Discrete or Continuous II Identify the follow- ing as discrete or continuous random variables:a. Increase in length of life attained by a cancer patient as a result of surgeryb. Tensile breaking strength (in kilograms per square centimetre) of 2.5-centimetre-diameter steel cablec. Number of
4.83 Discrete or Continuous? Identify the follow- ing as discrete or continuous random variables:a. Total number of points scored in a football gameb. Shelf life of a particular drugc. Height of the ocean's tide at a given locationd. Length of a two-year-old black basse. Number of aircraft
Determine the yearly premium for a $1000 insurance policy covering an event that, over a long period of time, has occurred at the rate of 2 times in 100. Let X equal the yearly financial gain to the insurance company resulting from the sale of the policy, and let C equal the unknown yearly premium.
In a lottery conducted to benefit the local fire station, 8000 tickets are to be sold at $5 each. The prize is $12,000. If you purchase two tickets, what is your expected gain?
An electronics store sells a particular model of computer laptop. There are only four laptops in stock, and the manager wonders what today's demand for this particular model will be. She learns from the marketing department that the probability distribu- tion for X, the daily demand for the laptop,
Toss two fair coins and let X equal the number of heads observed. Find the probability distribution for X.
4.82 Screening Tests Suppose that a certain disease is present in 10% of the population, and that there is a screening test designed to detect this disease if present. The test does not always work perfectly. Sometimes the test is negative when the disease is present, and sometimes it is positive
4.81 Cheating on Your Taxes? Suppose 5% of all people filing the income tax form seek deductions that they know are illegal, and an additional 2% incorrectly list deductions because they are unfamiliar with income tax regulations. Of the 5% who are guilty of cheating, 80% will deny knowledge of the
4.80 Medical Diagnostics Medical case histories indicate that different illnesses may produce identi- cal symptoms. Suppose a particular set of symptoms, which we will denote as event H, occurs only when any one of three illnesses A, B, or C-occurs. (For the sake of simplicity, we will assume that
4.79 No Pass, No Play Many public schools are implementing a "no pass, no play" rule for athletes. Under this system, a student who fails a course is disqualified from participating in extracurricular activities during the next grading period. Suppose the probability that an athlete who has not
4.78 Football Strategies A particular football team is known to run 30% of its plays to the left and 70% to the right. A linebacker on an opposing team notes that the right guard shifts his stance most of the time (80%) when plays go to the right and that he uses a balanced stance the remainder of
4.77 Airport Security Suppose that, in a particular city, airport A handles 50% of all airline traffic, and airports B and C handle 30% and 20%, respectively. The detection rates for weapons at the three airports are 0.9, 0.5, and 0.4, respectively. If a passenger at one of the airports is found to
4.76 Worker Error A worker-operated machine produces a defective item with probability 0.01 if the worker follows the machine's operating instructions exactly, and with probability 0.03 if the worker does not. If the worker follows the instructions 90% of the time, what proportion of all items
4.75 Violent Crime City crime records show that 20% of all crimes are violent and 80% are non-violent,involving theft, forgery, and so on. Ninety percent of violent crimes are reported versus 70% of non-violent crimes.a. What is the overall reporting rate for crimes in the city?b. If a crime in
4.74 Law of Total Probability A population can be divided into two subgroups that occur with probabilities 60% and 40%, respectively. An event A occurs 30% of the time in the first subgroup and 50% of the time in the second subgroup. What is the unconditional probability of the event A, regardless
4.73 Bayes' Rule II If an experiment is conducted, one and only one of three mutually exclusive events S1, S2, and S3 can occur, with these probabilities: P(S) = 0.2 P(S2) = 0.5 P(S3)=0.3 The probabilities of a fourth event A occurring, given that event S1, S2, or S3 occurs, are P(A|S) = 0.2
4.72 Bayes' Rule A sample is selected from one of two populations, S and S2, with probabilities P(S) = 0.7 and P(S2)=0.3. If the sample has been selected from S, the probability of observing an event A is P(AS) 0.2. Similarly, if the sample has been selected from S2, the probability of observing A
Refer to Example 4.24. Find the probability that the person selected is 45 years of age or older, and female, given that the individual is employed full-time.
What are the deal breakers for you on the dating scene? Several questions were asked of respondents in an online survey conducted by the Globe and Mail on Friday, June 25, 2009. More than 7,500 Canadian coast to coast participated in this poll. One of the questions was: Which of the following is
Refer to Project 3-A. Canada has a Road Safety Vision of having the safest roads in the world. Yet, the leading cause of death of Canadian children remains vehicle crashes. In 2006, a national child seat safety survey was conducted by an AUTO21 research team in collaboration with Transport Canada
The XL-2100 lamp unit is designed to be used with the Sony LCD projection TV "Grand WEGA." An electronics company purchases the lamp unit from three manu- factures located in Japan, China, and Korea respectively. The company located in Japan delivers the shipment on time 90% of the time, the
From the 2006 population estimates by Statistics Canada, Table 4.7 gives the fraction of Canadians 15 years of age and older who are employed full-time in each of three age groups by gender. The table also shows the fraction of Canadians 15 years of age and older in each age group. Use the Law of
4.71 Golfing Player A has entered a golf tournament but it is not certain whether player B will enter. Player A has a probability of 1/6 of winning the tournament if player B enters, and a probability of 3/4 of winning if player B does not enter the tournament. If the probabil- ity that player B
4.70 Jordan and Alex The top two 2006-2007 NHL shooting percentage leaders are Jordan Staal (Pittsburgh) and Alex Tanguay (Calgary). Canoe.ca reports that the shooting percentage for Jordan is 26.0, while for Alex it is 22.8. Assume that the shots are independent, and that each player takes two
4.69 Wage Losses and the "Motherhood Gap" According to a national survey by TD Economics, 81% of working mothers in Canada re-enter the workforce after leaving their jobs to have kids. Three moms were randomly selected:a. What is the probability that first selected mom has not re-entered the
4.68 Choosing a Mate Men and women often disagree on how they think about selecting a mate. Suppose that a poll of 1000 individuals in their twenties gave the following responses to the question of whether it is more important for their future mate to be able to communicate their feelings (F) than
4.67 Canadian Youth Survey This exercise uses a collection containing responses by Ontario youths (ages 10-13) to a survey as part of a long-term study conducted from 1994 through 1999.5 The question was asked: "How do you feel about school?" The table summarizes the results.If one individual is
4.66 Online Dating According to a BBC poll, nearly one-third of all Web users surf the Internet for a boyfriend or girlfriend. The BBC poll surveyed close to 11,000 Internet users in 19 countries. In Canada, according to another poll, only one-quarter of Internet users have tried online dating; 69%
4.65 Plant Genetics Gregor Mendel was a monk who suggested in 1865 a theory of inheritance based on the science of genetics. He identified heterozygous individuals for flower colour that had two alleles (one r recessive white colour allele and one R = dominant red colour allele). When these
4.64 Social Media and Canadians A Vancouver- based Internet marketing firm called 6S Marketing conducted a survey of its database to track the use of social media in Canada. They polled 10,000 Canadians and uncovered some interesting data: 70% of Canadians say they use social media. Facebook is the
4.63 Smoke Detectors A smoke-detector system uses two devices, A and B. If smoke is present, the probability that it will be detected by device A is 0.95; by device B, 0.98; and by both devices, 0.94.a. If smoke is present, find the probability that the smoke will be detected by device A or device
4.62 Canadians Are Huge Online Users According to comScore, a company that provides digital marketing intelligence, Canadians spend more time online when compared to a group of countries that includes the United States, France, and the United Kingdom. In Canada, 68% of the population is online, as
4.61 Smoking and Cancer A survey of people in a given region showed that 20% were smokers. The probability of death due to lung cancer, given that a person smoked, was roughly 10 times the probability of death due to lung cancer, given that a person did not smoke. If the probability of death due to
4.60 Inspection Lines A certain manufactured item is visually inspected by two different inspectors. When a defective item comes through the line, the probability that it gets by the first inspector is 0.1. Of those that get past the first inspector, the second inspector will "miss" 5 out of 10.
4.59 Alcohol Consumption and Smoking among Canadian Medical Students A team of researchers at the Centre for Addiction and Mental Health, Toronto, conducted a survey to quantify the extent, and to assess student perception, of alcohol and tobacco use among medical students at the University of
4.58 Starbucks or Tim Hortons? A university student frequents one of two coffee houses on campus, choosing Tim Hortons 70% of the time and Starbucks 30% of the time. Regardless of where she goes, she buys a decaffeinated coffee on 60% of her visits.a. The next time she goes into a coffee house on
4.57 The Birthday Problem, continued If n people enter a room, find these probabilities: A: None of the people have the same birthday B: At least two of the people have the same birthday Solve fora. n=3b. n=4 (NOTE: Surprisingly, P(B) increases rapidly as n increases. For example, for n = 20,
4.56 The Birthday Problem Two people enter a room and their birthdays (ignoring years) are recorded.a. Identify the nature of the simple events in S.b. What is the probability that the two people have a specific pair of birthdates?c. Identify the simple events in event A: Both people have the same
4.55 Drug Offenders, continued Use the prob- abilities of Exercise 4.54 to show that these equalities are true: =a. P(AB) P(A)P(B|A)b. P(AB) = P(B)P(A|B)c. P(AUB) = P(A) + P(B)-P(AB)
4.54 Drug Offenders A study of the behaviour of a large number of drug offenders after treatment for drug abuse suggests that the likelihood of conviction within a two-year period after treatment may depend on the offender's education. The proportions of the total number of cases that fall into

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